Maurice Pouzet
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Maurice Pouzet has a total of 55 co-authors. Here are the top co-authors by number of shared publications:
- Claude Laflamme
Department University of Calgary
9 shared publications - Robert Woodrow
Department University of Calgary
7 shared publications - Christian Delhommé
Université de la Réunion
6 shared publications - Hamza Si Kaddour
Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 Novembre 1918, F-69622 Villeurbanne cedex, France
6 shared publications - Norbert Sauer
Department of Mathematics and Statistics, University of Calgary, Calgary, Canada
6 shared publications - Imed Zaguia
Royal Mil Coll Canada, Dept Math & Comp Sci, POB 17000, Kingston, ON K7K 7B4, Canada
5 shared publications
Latest Publications
- 3 December 2025
- Published by Springer Nature in Mediterranean Journal of Mathematics
- Vol. 23 (1) , 26
Ordered Structures with No Finite Monomorphic Decomposition: Application to the Profile of Hereditary Classes
- 3 December 2025
- Published by Springer Nature in Mediterranean Journal of Mathematics
- Vol. 23 (1) , 26
Ordered Structures with No Finite Monomorphic Decomposition: Application to the Profile of Hereditary Classes
- Open Access
The structure of interval orders with no infinite antichain
We prove that every interval order $P$ with no infinite antichain has a Gallai decomposition. That is, $P$ is a lexicographical sum of proper interval orders over a chain, an antichain or a prime interval order. This is a consequence of the fact that the tree decomposition of a graph into robust modules, as introduced by Courcelle and Delhommé (Theoretical Computer Science \textbf{394} (2008) 1--38), is chain finite whenever the graph has no infinite independent sets. Next, we prove that every prime interval order with no infinite antichain is at most countable and scattered. Furthermore, for each countable ordinal $α$ we exhibit an example of a well-quasi-ordered prime interval order $P_α$ whose chain of maximal antichains has Hausdorff rank $α$.Open AccessThe structure of interval orders with no infinite antichain
We prove that every interval order $P$ with no infinite antichain has a Gallai decomposition. That is, $P$ is a lexicographical sum of proper interval orders over a chain, an antichain or a prime interval order. This is a consequence of the fact that the tree decomposition of a graph into robust modules, as introduced by Courcelle and Delhommé (Theoretical Computer Science \textbf{394} (2008) 1--38), is chain finite whenever the graph has no infinite independent sets. Next, we prove that every prime interval order with no infinite antichain is at most countable and scattered. Furthermore, for each countable ordinal $α$ we exhibit an example of a well-quasi-ordered prime interval order $P_α$ whose chain of maximal antichains has Hausdorff rank $α$. - 3 April 2024
- Published by Springer Nature in Order
- Vol. 41 (1) , 183-278
Well-quasi-ordering and Embeddability of Relational Structures
- 3 April 2024
- Published by Springer Nature in Order
- Vol. 41 (1) , 183-278
Well-quasi-ordering and Embeddability of Relational Structures
- Open Access
Absolute retracts of reflexive oriented graphs: the role of the MacNeille completion
We characterize the absolute retracts in the category of reflexive oriented graphs, that is, antisymmetric reflexive graphs, where morphisms between objects preserve arcs (which may be sent to loops). Here we show, by correcting a much earlier attempt at a proof, that a reflexive oriented graph is an absolute retract if and only if it is indeed a retract of some (direct) product of reflexive oriented zigzags (which are concatenations of reflexive oriented paths). Absolute retracts are therefore necessarily acyclic. In contrast to other categories of graphs and ordered sets, not every acyclic oriented graph can be embedded isometrically into some absolute retract. Embedding involves isometry with respect to the zig-zag distances forming a particular "dual quantale", which is a complete lattice of certain sets of words over the alphabet $\{+, -\}$, endowed with an additional monoid operation (viz., compound concatenation of sets of words) and an involution (interchanging $+$ and $- $ and then mirroring words). As reflexive oriented zigzags have MacNeille-closed distances, so do their products and retracts. So, the category of reflexive oriented graphs and its full subcategory of reflexive acyclic graphs do not have enough injectives, as the injective objects coincide with the absolute retracts.Open AccessAbsolute retracts of reflexive oriented graphs: the role of the MacNeille completion
We characterize the absolute retracts in the category of reflexive oriented graphs, that is, antisymmetric reflexive graphs, where morphisms between objects preserve arcs (which may be sent to loops). Here we show, by correcting a much earlier attempt at a proof, that a reflexive oriented graph is an absolute retract if and only if it is indeed a retract of some (direct) product of reflexive oriented zigzags (which are concatenations of reflexive oriented paths). Absolute retracts are therefore necessarily acyclic. In contrast to other categories of graphs and ordered sets, not every acyclic oriented graph can be embedded isometrically into some absolute retract. Embedding involves isometry with respect to the zig-zag distances forming a particular "dual quantale", which is a complete lattice of certain sets of words over the alphabet $\{+, -\}$, endowed with an additional monoid operation (viz., compound concatenation of sets of words) and an involution (interchanging $+$ and $- $ and then mirroring words). As reflexive oriented zigzags have MacNeille-closed distances, so do their products and retracts. So, the category of reflexive oriented graphs and its full subcategory of reflexive acyclic graphs do not have enough injectives, as the injective objects coincide with the absolute retracts. - Open Access
Ordered structures with no finite monomorphic decomposition. Application to the profile of hereditary classes
We present a structural approach of some results about jumps in the behavior of the profile (alias generating function) of hereditary classes of finite structures. We consider the following notion due to N.Thiéry and the second author. A \emph{monomorphic decomposition} of a relational structure $R$ is a partition of its domain $V(R)$ into a family of sets $(V_x)_{x\in X}$ such that the restrictions of $R$ to two finite subsets $A$ and $A'$ of $V(R)$ are isomorphic provided that the traces $A\cap V_x$ and $A'\cap V_x$ have the same size for each $x\in X$. Let $\mathscr S_μ$ be the class of relational structures of signature $μ$ which do not have a finite monomorphic decomposition. We show that if a hereditary subclass $\mathscr D$ of $\mathscr S_μ$ is made of ordered relational structures then it contains a finite subset $\mathfrak A$ such that every member of $\mathscr D$ embeds some member of $\mathfrak A$. Furthermore, for each $R\in \mathfrak A$ the profile of the age $\age(R)$ of $R$ (made of finite substructures of $R$) is at least exponential. We deduce that if the profile of a hereditary class of finite ordered structures is not bounded above by a polynomial then it is at least exponential. For ordered graphs, this result is a part of classification obtained by Balogh, Bollobás and Morris (2006).Open AccessOrdered structures with no finite monomorphic decomposition. Application to the profile of hereditary classes
We present a structural approach of some results about jumps in the behavior of the profile (alias generating function) of hereditary classes of finite structures. We consider the following notion due to N.Thiéry and the second author. A \emph{monomorphic decomposition} of a relational structure $R$ is a partition of its domain $V(R)$ into a family of sets $(V_x)_{x\in X}$ such that the restrictions of $R$ to two finite subsets $A$ and $A'$ of $V(R)$ are isomorphic provided that the traces $A\cap V_x$ and $A'\cap V_x$ have the same size for each $x\in X$. Let $\mathscr S_μ$ be the class of relational structures of signature $μ$ which do not have a finite monomorphic decomposition. We show that if a hereditary subclass $\mathscr D$ of $\mathscr S_μ$ is made of ordered relational structures then it contains a finite subset $\mathfrak A$ such that every member of $\mathscr D$ embeds some member of $\mathfrak A$. Furthermore, for each $R\in \mathfrak A$ the profile of the age $\age(R)$ of $R$ (made of finite substructures of $R$) is at least exponential. We deduce that if the profile of a hereditary class of finite ordered structures is not bounded above by a polynomial then it is at least exponential. For ordered graphs, this result is a part of classification obtained by Balogh, Bollobás and Morris (2006). - 3 October 2023
- Published by Elsevier in European Journal of Combinatorics
- Vol. 119
Hereditary classes of ordered sets of width at most two
- 3 October 2023
- Published by Elsevier in European Journal of Combinatorics
- Vol. 119
Hereditary classes of ordered sets of width at most two
- Open Access
A few characterizations of topological spaces with no infinite discrete subspace
We give several characteristic properties of FAC spaces, namely topological spaces with no infinite discrete subspace. The first one was obtained in 2019 by the first author, and states that every closed set is a finite union of irreducible closed subsets. The full result extends well-known characterizations of posets with no infinite antichain. One of them is that FAC spaces are, equivalently, topological spaces in which every closed set contains a dense Noetherian subspace, or spaces in which every Hausdorff subspace is finite, or in which no subspace has any infinite relatively Hausdorff subset. The latter comes with a nice min-max property, extending an observation of Erd\"os and Tarski in the case of posets: on spaces with no infinite relatively Hausdorff subset, the cardinalities of relatively Hausdorff subsets are bounded, and the least upper bound is also the least cardinality of a family of closed irreducible subsets that cover the space.
Open AccessA few characterizations of topological spaces with no infinite discrete subspace
We give several characteristic properties of FAC spaces, namely topological spaces with no infinite discrete subspace. The first one was obtained in 2019 by the first author, and states that every closed set is a finite union of irreducible closed subsets. The full result extends well-known characterizations of posets with no infinite antichain. One of them is that FAC spaces are, equivalently, topological spaces in which every closed set contains a dense Noetherian subspace, or spaces in which every Hausdorff subspace is finite, or in which no subspace has any infinite relatively Hausdorff subset. The latter comes with a nice min-max property, extending an observation of Erd\"os and Tarski in the case of posets: on spaces with no infinite relatively Hausdorff subset, the cardinalities of relatively Hausdorff subsets are bounded, and the least upper bound is also the least cardinality of a family of closed irreducible subsets that cover the space.
- Open Access
The chain covering number of a poset with no infinite antichains
The chain covering number $\Cov(P)$ of a poset $P$ is the least number of chains needed to cover $P$. For a cardinal $\nu$, we give a list of posets of cardinality and covering number $\nu$ such that for every poset $P$ with no infinite antichain, $\Cov(P)\geq \nu$ if and only if $P$ embeds a member of the list. This list has two elements if $\nu$ is a successor cardinal, namely $[\nu]^2$ and its dual, and four elements if $\nu$ is a limit cardinal with $\cf(\nu)$ weakly compact. For $\nu= \aleph_1$, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal $\nu$.Open AccessThe chain covering number of a poset with no infinite antichains
The chain covering number $\Cov(P)$ of a poset $P$ is the least number of chains needed to cover $P$. For a cardinal $\nu$, we give a list of posets of cardinality and covering number $\nu$ such that for every poset $P$ with no infinite antichain, $\Cov(P)\geq \nu$ if and only if $P$ embeds a member of the list. This list has two elements if $\nu$ is a successor cardinal, namely $[\nu]^2$ and its dual, and four elements if $\nu$ is a limit cardinal with $\cf(\nu)$ weakly compact. For $\nu= \aleph_1$, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal $\nu$. - Open Access
Hereditary classes of ordered sets of width at most two
This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs. Several open problems about hereditary classes of relational structures which have been considered over the years have positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ru\v{s}kuc and Vatter. We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver's theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo. Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added.
Open AccessHereditary classes of ordered sets of width at most two
This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs. Several open problems about hereditary classes of relational structures which have been considered over the years have positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ru\v{s}kuc and Vatter. We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver's theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo. Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added.
- Open Access
- 18 May 2022
- Published by University of Calgary in Contributions to Discrete Mathematics
- Vol. 17 (1) , 70-108
Some relational structures with polynomial growth and their associated algebras II. Finite generation.
The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every nonnegative integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $\mathbb{K}.\mathcal A$ introduced by P. J. Cameron. In a previous paper, we studied the relationship between the properties of a relational structure $R$ and those of its age algebra, particularly when $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups or the rings of quasisymmetric polynomials. The main theorem of this paper characterizes combinatorially when the age algebra is finitely generated in this setting. For tournaments, this boils down to the profile being bounded. We further investigate how far the well known algebraic properties of invariant rings and quasisymmetric polynomials extend to age algebras; notably, we explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra. For a homogeneous structure with a profile bounded by a polynomial, Cameron conjectured in the early eighties that the profile is asymptotically polynomial; Macpherson further conjectured that the age algebra is finitely generated. This was proven recently by Falque and the second author. The combined results support the conjecture that---assuming finite kernel---profiles bounded by a polynomial are asymptotically polynomial, and give hope for a complete characterization of when the age algebra is finitely generated.- 18 May 2022
- Published by University of Calgary in Contributions to Discrete Mathematics
- Vol. 17 (1) , 70-108
Open AccessSome relational structures with polynomial growth and their associated algebras II. Finite generation.
The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every nonnegative integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $\mathbb{K}.\mathcal A$ introduced by P. J. Cameron. In a previous paper, we studied the relationship between the properties of a relational structure $R$ and those of its age algebra, particularly when $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups or the rings of quasisymmetric polynomials. The main theorem of this paper characterizes combinatorially when the age algebra is finitely generated in this setting. For tournaments, this boils down to the profile being bounded. We further investigate how far the well known algebraic properties of invariant rings and quasisymmetric polynomials extend to age algebras; notably, we explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra. For a homogeneous structure with a profile bounded by a polynomial, Cameron conjectured in the early eighties that the profile is asymptotically polynomial; Macpherson further conjectured that the age algebra is finitely generated. This was proven recently by Falque and the second author. The combined results support the conjecture that---assuming finite kernel---profiles bounded by a polynomial are asymptotically polynomial, and give hope for a complete characterization of when the age algebra is finitely generated.